Enhancing utility and diversifying model risk in a portfolio optimization framework

ABSTRACT

A portfolio optimization process that diversifies model risk by favoring a more diversified portfolio over other portfolios with similar characteristics is provided. According to one aspect of the present invention, an intelligent search is performed for a diverse portfolio that meets a predetermined diversity budget. An initial portfolio is determined based upon an available set of financial products. The cost associated with more diversified portfolios compared to the initial portfolio is considered and one of the more diversified portfolios is selected that has an associated cost that is less than or equal to the predetermined diversity budget.

[0001] This is a continuation-in-part of application serial No.09/151,715, filed on Sep. 11, 1998, that is currently pending.

COPYRIGHT NOTICE

[0002] Contained herein is material that is subject to copyrightprotection. The copyright owner has no objection to the facsimilereproduction of the patent disclosure by any person as it appears in thePatent and Trademark Office patent files or records, but otherwisereserves all rights to the copyright whatsoever.

FIELD OF THE INVENTION

[0003] The invention relates generally to the field of financialadvisory services. More particularly, the invention relates to aportfolio optimization process that diversifies model risk by favoring amore diversified portfolio over other portfolios with similarcharacteristics.

BACKGROUND OF THE INVENTION

[0004] From a set of N financial products (N>1), an infinite number ofportfolios are available for investment. Existing computer financialanalysis systems (also referred to as “portfolio optimizers”) purport tohelp individuals select portfolios to meet their needs. These systemstypically implement mathematical models based upon standard optimizationtechniques involving mean-variance optimization theory. According to themean-variance approach to portfolio selection, an optimal portfolio offinancial products may be identified with reference to an investor'spreference for various combinations of risk and return and the set ofefficient portfolios (also referred to as the efficient set or theefficient frontier). FIG. 1 illustrates a feasible set of portfoliosthat represents all the portfolios that maybe formed from a particularset of financial products. The arc AC represents an efficient set ofportfolios that each provide the highest expected return for a givenlevel of risk. A portfolio's risk is typically measured by the standarddeviation of returns. In general, there are many portfolios that havealmost the same expected return and about the same level of risk as anyefficient portfolio (e.g., portfolio B and portfolio E). Sincestatistical estimates of expected returns and risk are used to calculateefficient portfolios, the calculated efficient set could deviate fromthe true efficient set. When “model risk” is considered, portfolios inan error space surrounding an optimal portfolio are virtuallyindistinguishable. By “model risk,” what is meant is theuncertainty/risk in the mathematical models employed and errors that maybe introduced when estimating the properties of the financial productsbased upon historical data which may contain inaccuracies, such asstatistical noise or measurement error, for example. An example of aproblem induced by measurement error is the potential for highlyconcentrated estimated efficient portfolios. For instance, consider anasset that has a large positive error in its expected return estimate.Efficient portfolios constructed ignoring the possibility of this largepositive error may yield portfolios with highly concentrated positionsin this asset.

[0005] Existing portfolio optimizers typically ignore model risk, likelybecause of the great amount of processing that is thought to be requiredto identify and select from the many indistinguishable portfolios. Priorart portfolio optimizers are notorious for recommending portfolios thathave counterintuitive properties, such as highly concentrated positionsin individual assets or asset classes. For example, the typicalportfolio optimizer, having ignored portfolio E because it is not in theefficient set, would suggest portfolio B which may include highlyconcentrated holdings in one of the underlying N assets. Suchrecommendations make users skeptical of the results of traditionalportfolio optimizers and discourage adoption of such investment tools.

[0006] One way investment managers have traditionally attempted tocompensate for the inadequacies of portfolio optimizers is by imposingconstraints or bounds on the optimizer in one or more dimensions. Forexample, an investment manager may limit exposures to certain assetclasses, limit short positions, etc. While these manual constraints canbe implemented with knowledge of the bounded universe from which theportfolio will ultimately be built, they have several limitations.First, these manual techniques do not take the cost of imposingconstraints on the optimization process into account. Additionally,manual solutions are typically only practical when the universe fromwhich the portfolio can be drawn is limited to one set of mutual funds,asset classes, or financial products.

[0007] In view of the foregoing, what is needed is a generalizedportfolio diversification approach that produces recommended portfoliosthat take into account inherent model risk and with which users will beintuitively comfortable, thereby fostering the adoption of optimizationtools. Additionally, rather than arbitrarily spreading assets out, it isdesirable for the decision to pursue more diversity in a portfolio toconsider the cost of such diversity, in terms of its effect on expectedreturn, risk, and/or utility, for example. Finally, it would beadvantageous for the diversification approach to be broadly applicableto the universe of financial products.

SUMMARY OF THE INVENTION

[0008] A portfolio optimization process that diversifies model risk byfavoring a more diversified portfolio over other portfolios with similarcharacteristics is described. Broadly stated the present inventioninvolves determining an initial portfolio, performing diversificationprocessing to identify one or more alternative portfolios havingincreased diversification, and selecting a recommended portfolio fromthe initial portfolio or the one or more alternative portfolios basedupon a set of one or more criteria.

[0009] According to one aspect of the present invention, a more diverseportfolio may be selected over an initial portfolio in order todiversify model risk with reference to a predetermined diversity budget.An initial portfolio of financial products is determined from anavailable set of financial products. One or more dimensions of an errorspace are searched for an alternate portfolio that is more diverse thanthe initial portfolio. A cost associated with the alternate portfolio isthen calculated by comparing the difference between a characteristic ofthe initial portfolio and a corresponding characteristic of thealternate portfolio. Finally, the alternate portfolio is selected as therecommended portfolio if the cost is less than or equal to thepredetermined diversity budget.

[0010] According to another aspect of the present invention anintelligent search is performed for a diverse portfolio that meets apredetermined diversity budget. An initial portfolio is determined basedupon an available set of financial products. The cost associated withmore diversified portfolios compared to the initial portfolio isconsidered and one of the more diversified portfolios is selected thathas an associated cost that is less than or equal to the predetermineddiversity budget.

[0011] Other features of the present invention will be apparent from theaccompanying drawings and from the detailed description which follows.

BRIEF DESCRIPTION OF THE DRAWINGS

[0012] The present invention is illustrated by way of example, and notby way of limitation, in the figures of the accompanying drawings and inwhich like reference numerals refer to similar elements and in which:

[0013]FIG. 1 illustrates a feasible set of portfolios that can be formedfrom a set of financial products.

[0014]FIG. 2 illustrates a financial advisory system according to oneembodiment of the present invention.

[0015]FIG. 3 is an example of a computer system upon which oneembodiment of the present invention may be implemented.

[0016]FIG. 4 is a simplified block diagram illustrating one embodimentof a financial analysis system that may employ the diversificationmechanism of the present invention.

[0017]FIG. 5 is a flow diagram illustrating portfolio optimizationprocessing according to one embodiment of the present invention.

[0018]FIG. 6 is a flow diagram illustrating diversification processingaccording to one embodiment of the present invention.

[0019]FIG. 7 is a flow diagram illustrating diversification processingaccording to another embodiment of the present invention.

[0020]FIG. 8 is a flow diagram illustrating the generation of a morediverse portfolio according to one embodiment of the present invention.

[0021]FIG. 9A illustrates an initially identified optimal portfolio.

[0022]FIG. 9B illustrates the effect of a maximum exposure constraint onthe portfolio of FIG. 9A.

[0023]FIG. 9C illustrates a diversified portfolio after one or morestopping conditions have been achieved.

[0024]FIG. 10 conceptually illustrates an approach for quickly finding adiversified portfolio that meets the diversity budget according to oneembodiment of the present invention.

DETAILED DESCRIPTION

[0025] A mechanism is described for diversifying model risk. Suchuncertainty/risk is inherent in the mathematical models and thehistorical data employed by portfolio optimizers, for example. Thediversification mechanism described herein may efficiently search anerror space proximate to an initially identified optimal portfolio foralternative portfolios that are more diverse than the initial portfolioand that are not too costly to implement in terms of differences inexpected returns, risk and/or utility. According to embodiments of thepresent invention, after an initial efficient portfolio is identified byan optimization process, various characteristics of the initialportfolio may be used as a baseline by a diversification process tomeasure the cost of implementing more diverse portfolios having verysimilar expected return, risk, and/or utility characteristics as theinitial portfolio. The more diverse portfolios may be located bysearching various dimensions of an error space that is proximate to theinitial portfolio. For example, the more diverse portfolios may beselected from a group of portfolios that have approximately the samelevel of risk and slightly lower expected returns than the initialportfolio or from a group of portfolios that have approximately the sameexpected returns but have a higher level of risk than the initialportfolio. In one embodiment, the diversification process favors morediverse portfolios over other portfolios with similar expected returncharacteristics by allocating a predetermined cost (referred to as thediversity budget) that can be spent in pursuit of diversity. In thismanner, of the portfolios that are evaluated in a predefined errorspace, the most diverse portfolio that stays within the diversity budgetwill be selected. In other embodiments, other stopping conditions mayalso be employed to terminate the diversity processing. For example, thesearch for a more diverse portfolio than the current portfolio may stopwhen, among other things: (1) maintaining certain desirablecharacteristics of the initial portfolio constant is no longer feasible;(2) the number of financial products in the current portfolio exceeds apredetermined number of financial products; and/or (3) a certain numberof iterations have been performed and/or a certain number of alternateportfolios have been considered.

[0026] In the following description, for the purposes of explanation,numerous specific details are set forth in order to provide a thoroughunderstanding of the present invention. It will be apparent, however, toone skilled in the art that the present invention may be practicedwithout some of these specific details. In other instances, well-knownstructures and devices are shown in block diagram form.

[0027] The present invention includes various steps, which will bedescribed below. The steps of the present invention may be embodied inmachine-executable instructions. The instructions can be used to cause ageneral-purpose or special-purpose processor which is programmed withthe instructions to perform the steps of the present invention.Alternatively, the steps of the present invention may be performed byspecific hardware components that contain hardwired logic for performingthe steps, or by any combination of programmed computer components andcustom hardware components.

[0028] The present invention may be provided as a computer programproduct which may include a machine-readable medium having storedthereon instructions which may be used to program a computer (or otherelectronic devices) to perform a process according to the presentinvention. The machine-readable medium may include, but is not limitedto, floppy diskettes, optical disks, CD-ROMs, and magneto-optical disks,ROMs, RAMs, EPROMs, EEPROMs, magnetic or optical cards, or other type ofmedia/machine-readable medium suitable for storing electronicinstructions. Moreover, the present invention may also be downloaded asa computer program product, wherein the program may be transferred froma remote computer (e.g., a server) to a requesting computer (e.g., aclient) by way of data signals embodied in a carrier wave or otherpropagation medium via a communication link (e.g., a modem or networkconnection).

[0029] While, embodiments of the present invention will be describedwith reference to a financial advisory system, the method and apparatusdescribed herein are equally applicable to other types of assetallocation applications, financial planning applications, investmentadvisory services, and financial product selection services, automatedfinancial product screening tools such as electronic personal shoppingagents and the like.

System Overview

[0030] The present invention may be included within a client-serverbased financial advisory system 200 such as that illustrated in FIG. 2.According to the embodiment depicted in FIG. 2, the financial advisorysystem 200 includes a financial staging server 220, a broadcast server215, a content server 217, an AdviceServer™ 210 (AdviceServer is atrademark of Financial Engines, Inc., the assignee of the presentinvention), and a client 205.

[0031] The financial staging server 220 may serve as a primary stagingand validation area for the publication of financial content. In thismanner, the financial staging server 220 acts as a data warehouse. Rawsource data, typically time series data, may be refined and processedinto analytically useful data on the financial staging server 220. On amonthly basis, or whatever the batch processing interval may be, thefinancial staging server 220 converts raw time series data obtained fromdata vendors from the specific vendor's format into a standard formatthat can be used throughout the financial advisory system 200. Variousfinancial engines may also be run to generate data for validation andquality assurance of the data received from the vendors. Anycalibrations of the analytic data needed by the financial engines may beperformed prior to publishing the final analytic data to the broadcastserver 215.

[0032] The broadcast server 215 is a database server. As such, it runsan instance of a Relational Database Management System (RDBMS), such asMicrosoft™ SQL-Server, Oracle™ or the like. The broadcast server 215provides a single point of access to all fund information and analyticdata. When advice servers such as AdviceServer 210 need data, they mayquery information from the broadcast server database. The broadcastserver 215 may also populate content servers, such as content server217, so remote implementations of the AdviceServer 210 need notcommunicate directly with the broadcast server 215. The AdviceServer 210is the primary provider of services for the client 205. The AdviceServer210 also acts as a proxy between external systems, such as externalsystem 225, and the broadcast server 215 or the content server 217.

[0033] According to the embodiment depicted, the user may interact withand receive feedback from the financial advisory system 200 using clientsoftware which may be running within a browser application or as astandalone desktop application on the user's personal computer 205. Theclient software communicates with the AdviceServer 210 which acts as aHTTP server.

An Exemplary Computer System

[0034] Having briefly described an exemplary financial advisory system200 which may employ various features of the present invention, acomputer system 300 representing an exemplary client 105 or server inwhich features of the present invention may be implemented will now bedescribed with reference to FIG. 3. Computer system 300 comprises a busor other communication means 301 for communicating information, and aprocessing means such as processor 302 coupled with bus 301 forprocessing information. Computer system 300 further comprises a randomaccess memory (RAM) or other dynamic storage device 304 (referred to asmain memory), coupled to bus 301 for storing information andinstructions to be executed by processor 302. Main memory 304 also maybe used for storing temporary variables or other intermediateinformation during execution of instructions by processor 302. Computersystem 300 also comprises a read only memory (ROM) and/or other staticstorage device 306 coupled to bus 301 for storing static information andinstructions for processor 302.

[0035] A data storage device 307 such as a magnetic disk or optical discand its corresponding drive may also be coupled to computer system 300for storing information and instructions. Computer system 300 can alsobe coupled via bus 301 to a display device 321, such as a cathode raytube (CRT) or Liquid Crystal Display (LCD), for displaying informationto a computer user. For example, graphical depictions of expectedportfolio performance, asset allocation for an optimal portfolio, chartsindicating short- and long-term financial risk, icons indicative of theprobability of achieving various financial goals, and other data typesmay be presented to the user on the display device 321. Typically, analphanumeric input device 322, including alphanumeric and other keys, iscoupled to bus 301 for communicating information and/or commandselections to processor 302. Another type of user input device is cursorcontrol 323, such as a mouse, a trackball, or cursor direction keys forcommunicating direction information and command selections to processor302 and for controlling cursor movement on display 321.

[0036] A communication device 325 is also coupled to bus 301 foraccessing remote servers, such as the AdviceServer 210, or other serversvia the Internet, for example. The communication device 325 may includea modem, a network interface card, or other well known interfacedevices, such as those used for coupling to Ethernet, token ring, orother types of networks. In any event, in this manner, the computersystem 300 may be coupled to a number of servers via a conventionalnetwork infrastructure, such as a company's Intranet and/or theInternet, for example.

Exemplary Financial Analisis System

[0037]FIG. 4 is a simplified block diagram illustrating a financialanalysis system 400 in which one embodiment of the present invention maybe used. Generally, the financial advisory system 400 includes asimulation module 440, a portfolio optimization module 456, and a userinterface (UI) 460. The UI 460 may include various mechanisms for datainput and output to provide the user with a means of interacting withand receiving feedback from the financial advisory system 400,respectively. Both the simulation module 440 and the portfoliooptimization module may receive input data from the user interface (UI)460 and provide data, such as financial products' exposures to variousfactors, probability distributions, and recommended portfolios offinancial products, to the UI 460.

[0038] The simulation module 440 may include a simulation engine forempirically generating draws from a random distribution. According tothe embodiment depicted, the simulation module 440 further includes apricing module 410, a factor module 420, and a style analysis module430.

[0039] The pricing module 410 may generate pricing data for one or moreassets. In one embodiment, pricing module 410 generates pricing data forthree assets (e.g., short-term bonds, long-term bonds and U.S.equities). These assets are used as core assets by simulation module 440for simulation functions. Alternatively, the core assets may bedifferent types of assets, such as U.S. equities and bonds (making nodistinction between short-term and long-term bonds). Of course, adifferent number of core assets may also be used.

[0040] In one embodiment, pricing module 410 generates a number of assetscenarios. Each scenario is an equally likely outcome based on theinputs to financial advisory system 400. By generating a number ofscenarios with pricing module 410, financial advisory system 400 maygenerate statistics for different projected asset valuations. Forexample, financial advisory system 400 may provide probabilitydistributions for each projected asset valuation.

[0041] Factor module 420 receives core asset pricing data from pricingmodule 410 and maps the data onto a set of factors. Factors output byfactor module 420 are used by returns-based style analysis module 430 togenerate style exposures for particular assets. Factor modules and styleanalysis are well known in the art and are not described in greaterdetail herein. Factor module 420 and style analysis module 430 mayperform the functions as described in “Asset allocation: Managementstyle and performance measurement,” by William F. Sharpe, Journal ofPortfolio Management, Vol. 18, No. 2, which is hereby incorporated byreference.

[0042] The portfolio optimization module 456 may determine optimalportfolios based on input provided to financial advisory system 400 viaUI 460. In the embodiment depicted, the portfolio optimization module456 further comprises a diversification module 455 and an optimizationmodule 450. The optimization module 450 calculates the utilitymaximizing set of financial products under a set of constraints definedby the user and the available feasible investment set. In oneembodiment, the calculation is based upon a mean-variance optimizationof the financial products.

[0043] The diversification module 455 manages diversification processingand evaluates the cost of performing diversification. As will bedescribed further below, during diversification processing, thediversification module 455 may cause the optimization module 450 toperform several iterations of optimization processing with variousconstraints, such as a maximum exposure to any individual financialproduct and/or a minimum exposure to any individual financial product.In one embodiment, the diversity budget is set to an appropriate defaultlevel. The appropriate default level may be determined by tuning aparameter utilized the financial analysis system until satisfactoryresults are achieved, for example. In another embodiment, the user mayprovide a preference for diversification via the UI 460, which may inturn be used to determine the diversity budget. Depending upon theuser's expressed preference for diversity, a diversity budget, typicallyfrom 0 basis points to 16 basis points may be allocated, for example,corresponding to a preference for no diversity and a high preference fordiversity, respectively. Importantly, as will be discussed furtherbelow, rather than arbitrarily spreading assets out, the decision topursue more diversity in a portfolio by the diversification module 455is made after explicitly considering cost of such diversity, in terms ofits effect on expected return, risk, and/or utility, for example.

[0044] Importantly, the portfolio optimization module 456 may execute ona server or on the same computer upon which the UI 460 resides.

[0045] Further description of a financial advisory system that mayincorporate various features of the present invention is disclosed in acopending U.S. patent application entitled “USER INTERFACE FOR AFINANCIAL ADVISORY SYSTEM,” application Ser. No. 09/904,707, filed onJul. 12, 2001 that is assigned to the assignee of the present inventionand which is hereby incorporated by reference.

Portfolio Optimization

[0046] In general, portfolio optimization is the process of determininga set of financial products that maximizes the utility function of auser. According to one embodiment, portfolio optimization processingassumes that users have a mean-variant utility function, namely, thatpeople like having more expected wealth and dislike volatility ofwealth. Based on this assumption, given a user's risk tolerance, theportfolio optimization module 456 may calculate an initial mean-varianceefficient optimal portfolio from a set of financial products that areavailable to the user. Depending upon the user's diversity preference,other more diversified portfolios may then be considered for purposes ofdiversifying model risk. Preferably, both the optimization problem andthe diversification problem are expressed as a series of one or moreQuadratic Programming (QP) problems. QP is a technique for solvingoptimization problems involving quadratic (squared terms) objectivefunctions with linear equality and/or inequality constraints. A numberof different QP techniques exist, each with different properties. Forexample, some are better for suited for small problems, while others arebetter suited for large problems. Some are better for problems with veryfew constraints and some are better for problems with a large number ofconstraints. According to one embodiment of the present invention, whenQP is called for, an approach referred to as an “active set” method isemployed herein. The active set method is explained in Gill, Murray, andWright, “Practical Optimization,” Academic Press, 1981, Chapter 5.Advantageously, if the diversification problem can be structured as aseries of one or more QP problems, then interactive applications, suchas software that provides financial advice to individuals, may performdiversification processing in real-time.

[0047] Referring now to FIG. 5, portfolio optimization processingaccording to one embodiment of the present invention will now bedescribed. In one embodiment, the steps described below may be performedunder the control of a programmed processor, such as processor 302resident in client 205, or one of the servers 220, 215, 217, or 210. Atstep 510, an initial optimal portfolio is determined. According to oneembodiment of the present invention, the optimal portfolio is amean-variance efficient portfolio which may be determined with referenceto user-supplied data regarding his/her desirability for variouscombinations of risk and return. In this example, wealth in real dollarsmay be optimized by maximizing the following mean-variance utilityfunction by determining portfolio proportions (X_(i)): $\begin{matrix}{U = {{E(W)} - \frac{{Var}\quad (W)}{\tau}}} & \text{(EQ~~\#1)}\end{matrix}$

[0048] where for a given scenario,

[0049] E(W) is the expected value of wealth

[0050] Var(W) is the variance of wealth

[0051]₉₆ is the user's risk tolerance $\begin{matrix}{W = {W_{0}{\sum\limits_{i = 1}^{n}{X_{i}( {1 + R_{i}} )}}}} & {\text{(EQ~~\#2)}\quad}\end{matrix}$

[0052] where,

[0053] W₀=initial wealth

[0054] R_(l)=random return on financial product i

[0055] X_(l)=represents the recommended constant proportion of eachcontribution allocated to financial product i.

0≦X_(l)≦UB

[0056] UB=Upper bound on maximum exposure

[0057] n is the number of financial products that are available foroptimization.

[0058] Alternately, the following equation could be substituted for EQ#2: $\begin{matrix}\begin{matrix}{W_{T} = \quad {{X_{1}{\sum\limits_{t = 0}^{T - 1}{C_{t}{\prod\limits_{j = {t + 1}}^{T}\quad ( {1 + R_{j1}} )}}}} + \ldots +}} \\{\quad {{X_{n}{\sum\limits_{t = 0}^{T - 1}{C_{t}{\prod\limits_{j = {t + 1}}^{T}( {1 + R_{jn}} )}}}} + g}}\end{matrix} & \text{(EQ~~\#2A)}\end{matrix}$

[0059] where,

[0060] X_(i) represents the recommended constant proportion of each netcontribution that should be allocated to financial product i.

[0061] C_(t) represents the net contribution at time t,

[0062] R_(ji) represents the expected returns for financial product i inyear j,

[0063] n is the number of financial products that are available foroptimization,

[0064] g is the value of constrained assets for a given scenario,

[0065] The product of gross returns represents the compounding of valuesfrom year 1 to the horizon. Initial wealth in the portfolio isrepresented by contribution C₀.

[0066] Importantly, the financial product returns need not representfixed allocations of a single financial product. Within the context ofthe optimization problem, any individual asset return may be composed ofa static or dynamic strategy involving one or more financial products.For example, one of the assets may itself represent a constantre-balanced strategy over a group of financial products. Moreover, anydynamic strategy that can be formulated as an algorithm may beincorporated into the portfolio optimization. For example, an algorithmwhich specifies risk tolerance which decreases with the age of the usercould be implemented. It is also possible to incorporate path dependentalgorithms (e.g., portfolio insurance).

[0067] According to Equation #2A, contributions are made from thecurrent year to the year prior to retirement. Typically, a contributionmade at time t will be invested from time t until retirement. Anexception to this would be if a user specifies a withdrawal, in whichcase a portion of the contribution may only be held until the expectedwithdrawal date.

[0068] An alternative to the buy and hold investment strategy assumedabove would be to implement a “constant mix” investment strategy orre-balancing strategy. For purposes of this example, it is assumed thatthe recommended fixed target asset-mix will be held in an account foreach year in the future. Therefore, each year, assets will be boughtand/or sold to achieve the target. Let f_(l) be the fraction of accountwealth targeted for the i-th asset, then the sum of the fractions mustequal one.

[0069] In the following “evolution” equations, nominal wealthaggregation is modeled for a single taxable account from the currenttime t=0 to the time horizon t=T. It is assumed that “N” assets are inthe account, labeled by the set of subscripts {i=1, . . . , N}. Thesuperscripts minus and plus are used to distinguish between the valuesof a variable just before, and just after, “settlement”. The settlement“event” includes paying taxes on distributions and capital gains,investing new contributions, buying and selling assets to achieve theconstant mix, and paying load fees. For example, W⁺(t) is the totalwealth invested in all assets just after settlement at time “t”. Theevolution equations for the pre- and post-settlement values, the“dollars” actually invested in each asset, are: $\begin{matrix}{{W_{i}^{-}(t)} = \{ \begin{matrix}{{W_{i}^{-}(0)},} & {{t = 0},} \\{{{\lbrack {1 + {R_{i}(t)}} \rbrack \cdot {W_{i}^{+}( {t - 1} )}} - {{k_{i}(t)}}},} & {{0 < t \leq T},}\end{matrix} } & \text{(19a)} \\{{W_{i}^{+}(t)} = \{ \begin{matrix}{{f_{i} \cdot {W^{+}(t)}},} & {{0 \leq t < T},} \\{0,} & {t = {T.}}\end{matrix} } & \text{(19b)}\end{matrix}$

[0070] In the above equation, the double-bar operator ∥ ∥ is equal toeither its argument or zero, whichever is greater. From Eq.(19a), we seethat the pre-settlement value at any time (after the initial time) isjust the gross return on the post-settlement value of the previous timeless the “positive-part” of any distribution, i.e. the “dividend”. Here,k₁(t) is the portion of the return of the i-th asset that isdistributed, and R₁(t) is the total nominal return on the i-th asset inthe one-year period [t−1, t]. We also assume that an initial,pre-settlement value is given for each asset. Eq.(19b) defines thepost-settlement value in terms of the asset's constant mix and the totalaccount value after settlement. Since we “cash-out” the portfolio at thetime horizon, the final amount in each asset at t=T is zero. The pre-and post-settlement, total values are governed by the pair of equations:$\begin{matrix}{{{W^{-}(t)} = {\sum\limits_{i = 1}^{N}{W_{i}^{-}(t)}}},{0 \leq t \leq T},} & \text{(19c)}\end{matrix}$

[0071] In Eq.(19d), C(t) is the nominal contribution to the account attime “t”, D(t) is the total of all distributed “dividends”, L(t) is the“leakage”, the total amount paid in loads to both rebalance and toinvest additional contributions, and S(t) is the “shrinkage”, the totalamount paid in taxes on distributions and capital gains. We note thatW⁺(T) is the final horizon wealth after all taxes have been paid. Thevalue of D(t), the total of all distributed dividends, is the sum of thepositive distributions: $\begin{matrix}{{{D(t)} = {\sum\limits_{i = 1}^{N}{{k_{i}(t)}}}},{0 \leq t \leq {T.}}} & \text{(19e)}\end{matrix}$

[0072] Similarly, the “leakage” L(t) is the total amount of dollars paidin loads, and L_(l)(t) is the number of dollars paid in loads on justthe i-th asset. These individual loads depend on l_(i), the front-endload fee (a rate) on the i-th asset.

L _(l)(t)=[l _(i)/(1−l _(l))]·∥W _(i) ⁺(t)−∥k _(l)(t)∥−W _(i) ⁻(t)∥,0≦t≦T  (19f)

[0073] $\begin{matrix}{{{L(t)} = {\sum\limits_{i = 1}^{N}{L_{i}(t)}}},{0 \leq t \leq {T.}}} & \text{(19g)}\end{matrix}$

[0074] If there is a short-term loss (negative distribution), the loadfee paid on an asset's purchase is just a fixed fraction of the purchaseprice.^(i) When there is a short-term gain (positive distribution), wecan re-invest any part of it without load fees, and pay fees only onpurchases in excess of the gain. Note that at the horizon, we“cash-out”, and don't pay any load fees.

[0075] The equation for the “shrinkage” S(t), the total amount paid intaxes, has two terms. The first term is the tax on distributions and ismultiplied by the marginal tax-rate; the second term is the tax oncapital gains and is multiplied by the capital gains tax-rate.$\begin{matrix}{\begin{matrix}{{S(t)} = \quad {{\tau_{m} \cdot {\sum\limits_{i = 1}^{N}{k_{i}(t)}}} + {\tau_{cg} \cdot {\sum\limits_{i = 1}^{N}{\lbrack {1 - {{B_{i}( {t - 1} )}/{W_{i}^{-}(t)}}} \rbrack \cdot}}}}} \\{\quad {{{{W_{i}^{-}(t)} - {W_{i}^{+}(t)}}},}}\end{matrix}{0 \leq t \leq {T.}}} & \text{(19h)}\end{matrix}$

[0076] In Eq.(19h), the capital gains tax depends on the basis B_(l)(t),the total of all after-tax nominal-dollars that have been invested inthe i-th asset up to time “t”. Note that there can be either a capitalgain or loss. The double-bar operator ensures that capital gains aretriggered only when there is a sale of assets. At the horizon, we sellall assets, and automatically pay all taxes. The basis B_(i)(t), evolvesaccording to the following recursion equation: $\begin{matrix}{{B_{i}(t)} = \{ \begin{matrix}{{B_{i}(0)},} & {{t = 0},} \\{{B_{i}( {t - 1} )} + {{{W_{i}^{+}(t)} - {W_{i}^{-}(t)}}} + {L_{i}(t)}} & \quad \\{{{- \lbrack {{B_{i}( {t - 1} )}/{W_{i}^{-}(t)}} \rbrack} \cdot {{{W_{i}^{-}(t)} - {W_{i}^{+}(t)}}}},} & {0 < t \leq {T.}}\end{matrix} } & \text{(19i)}\end{matrix}$

[0077] Note that all new purchases are made with after-tax dollars, andadd to the basis; all sales decrease the basis. Further, any load paidto purchase an asset adds to the basis. We assume that the initial basisB₁(0) of an asset is either given, or defaults to the initial,pre-settlement value so that the average basis is initially equal toone.

[0078] A “constitutive” equation for k_(l)(t) is needed to complete oursystem of equations. Short-term distributions depend on the “type” ofasset; here we model mutual funds; $\begin{matrix}{{k_{i}(t)} = \{ \begin{matrix}{{k_{i}(0)},} & {{t = 0},} \\{{\kappa_{i} \cdot {R_{i}(t)} \cdot {W_{i}^{+}( {t - 1} )}},} & {0 < t \leq {T.}}\end{matrix} } & \text{(20a)}\end{matrix}$

[0079] Often, we set the initial distribution to zero, and assume thatthe asset's initial pre-settlement value has already accounted for anynon-zero, initial value. We note that the distribution is proportionalto the amount of wealth at “stake” during the prior-period. For mutualfunds, we assume that the distribution is a fraction κ₁ of theprior-period's total return, and therefore is also proportional toR₁(t). Note that the distribution in Eq.(20a) can be a gain (positive)or a loss (negative). In contrast, the constitutive equation for stockstakes the form: $\begin{matrix}{{k_{i}(t)} = \{ \begin{matrix}{{k_{i}(0)},} & {{t = 0},} \\{\quad {{\kappa_{i} \cdot \lbrack {1 + {R_{i}(t)}} \rbrack \cdot {W_{i}^{+}( {t - 1} )}},}} & {\quad {0 < t \leq {T.}}}\end{matrix} } & ( \text{20b} )\end{matrix}$

[0080] For stocks, the proportionality constant K₁ models a constantdividend “yield”, and the distribution is always a gain (non-negative).For stocks (mutual funds), the distribution is proportional to the gross(simple) return.

[0081] Before we leave this section, a word on 401(k) plans and IRA's(with no load funds). For these accounts, the loads and taxes areignored, and there is no basis in the asset. At “settlement”, the userjust re-balances their account. The evolution equations for theseaccounts is trivial in comparison to the equations for a general taxableaccount:

W _(i) ⁺(t)=f _(i) ·W ⁺(t), 0≦t≦T,  (21a)

[0082] $\begin{matrix}{{W^{+}(t)} = \{ \begin{matrix}{{W^{+}(0)},} & {{t = 0},} \\{\quad {{{( {1 + {\sum\limits_{i = 1}^{N}\quad {f_{i} \cdot {R_{i}(t)}}}} ) \cdot {W^{+}( {t - 1} )}} + {C(t)}},}} & {\quad {0 < t \leq {T.}}}\end{matrix}\quad } & \text{(21b)}\end{matrix}$

[0083] At the time horizon T, the total wealth in a non-taxable accountis just W⁺(T). This is a pre-withdrawal total value. When retirementwithdrawals are made from a tax-free account, they are taxed at theclient's average tax-rate, τ_(a). Therefore, the “after-tax” equivalentvalue is equal to “pre-tax” wealth W⁺(T) times the tax factor (1−τ_(a)).

[0084] How do we aggregate taxable and non-taxable accounts to get totalportfolio wealth? We choose non-taxable accounts as a baseline. If allthe funds in a non-taxable account were converted to an annuity, and theannuity payments were taken as withdrawals, then the withdrawals wouldmimic a salary subject to income taxes. This is precisely the client'spre-retirement situation. Before aggregating a taxable account, we scaleits “after-tax” value to this baseline using the formula:

W _(baseline) =W _(after-tax)/(1−τ_(a)).  (22)

[0085] Essentially, the baseline equivalent is obtained by grossing upvalues using the average tax-rate.

[0086] The evolution equation variables appear “implicitly” in therecursion relations. Hence, we need to “iterate” at each time step tosolve for “explicit” variable values.^(ii) We illustrate this processwith an example. Consider the simple case where there are nodistributions, contributions, or taxes; just loads, and a constant-mixstrategy. Here, the evolution equations simplify to a single equationfor the total, after-settlement wealth W⁺(t): $\begin{matrix}{{W^{+}(t)} = {{{W^{+}( {t - 1} )} \cdot {\sum\limits_{i = 1}^{N}\quad {f_{i} \cdot \lbrack {1 + {R_{i}(t)}} \rbrack}}} - {\sum\limits_{i = 1}^{N}\quad {f_{i} \cdot \lbrack {l_{i}/( {1 - l_{i}} )} \rbrack \cdot {{{{W^{+}(t)} - {\lbrack {1 + {R_{i}(t)}} \rbrack \cdot {W^{+}( {t - 1} )}}}}.}}}}} & (23)\end{matrix}$

[0087] Note, we only know W⁺(t) as an implicit function of W⁺(t−1), butgiven a guess for it's value, we can refine the guess by substituting itinto the right-side of Eq.(23).

[0088] It's instructive to re-write Eq.(23) as the pair of equations interms of an “effective” return R_(e)(t):

W ⁺(t)=[1+R _(e)(t)]·W ⁺(t−1),

[0089] $\begin{matrix}{{R_{e}(t)} = {{\sum\limits_{i = 1}^{N}\quad {f_{i} \cdot {R_{i}(t)}}} - {\sum\limits_{i = 1}^{N}\quad {f_{i} \cdot \lbrack {l_{i}/( {1 - l_{i}} )} \rbrack \cdot {{{{R_{e}(t)} - {R_{i}(t)}}}.}}}}} & \text{(24b)}\end{matrix}$

[0090] Eq.(24a) is the evolution equation for a single asset with theeffective return. Eq.(24b) is an implicit equation for the effectivereturn R_(e)(t) in terms of the asset returns R₁(t). We solve for theeffective return using iteration. When the loads are equal to zero, asexpected, the effective return is just a weighted-average of the assetreturns. Even when the loads are not zero, this average return is a goodinitial guess for the iteration procedure. In fact, using the averagereturn as the initial guess and iterating once yields the followingexplicit approximation for the effective return: $\begin{matrix}{{{R_{wgt}(t)} = {\sum\limits_{i = 1}^{N}\quad {f_{i} \cdot {R_{i}(t)}}}},} & \text{(25a)} \\{{R_{e}(t)} \approx {{R_{wgt}(t)} - {\sum\limits_{i = 1}^{N}\quad {f_{i} \cdot l_{i} \cdot {{{{R_{wgt}(t)} - {R_{i}(t)}}}.}}}}} & \text{(25b)}\end{matrix}$

[0091] Eq.(25b) is consistent with our intuition, and agrees well withhigher order iterates.

[0092] To determine the mutual fund input moments we must firstcalculate the kernel moments. This procedure calculates successiveannual kernel moments and averages the result. The resulting mean andcovariance matrix is then utilized by the reverse optimization procedureand also as an input into the optimization procedure.

[0093] To calculate analytic core moments, first we must describe thewealth for each core asset for an arbitrary holding period. For each ofthe core assets, the resulting wealth from an arbitrary investmenthorizon can be written as: [Note, this is an approximation for equities]$W_{t,T} = {\exp \{ {{\sum\limits_{j = t}^{T - 1}\quad a} + {b\quad X_{j + 1}} + {c{\prod\limits_{j + 1}{{+ d}\quad \delta_{j + 1}}}} + {e\quad X_{j}} + {f{\prod\limits_{j}{{+ g}\quad \delta_{j}}}}} \}}$

[0094] Where: a, b, c, d, e, f, g = Constants X_(j) = Real rate in yearj π_(j) = Inflation rate in year j δ_(j) = Dividend growth rate in yearj

[0095] The expectation of wealth for any of the core assets giveninformation at time zero is then:${E_{0}W_{t.T}} = {^{a{({T - t})}}E_{0}^{{\sum\limits_{j = t}^{T - 1}\quad {e\quad X_{j}}} + {b\quad X_{j + 1}}}E_{0}^{\sum\limits_{j = t}^{T - 1}\quad {f\prod\limits_{j + 1}}}E_{0}^{{\sum\limits_{j = t}^{T - 1}\quad {g\quad \delta_{j}}} + {d\quad \delta_{j + 1}}}}$

[0096] Since X, Π, and δ are independent, we can deal with each of theseexpectations separately. For example, consider the contribution in theabove equation from inflation. The summation can be rewritten as:${E_{0\quad}\exp \{ {\sum\limits_{j = t}^{T - 1}\quad {f{\prod\limits_{j}{{+ c}\prod\limits_{j + 1}}}}} \}} = {E_{0\quad}\exp \{ {{f{\prod\limits_{t}{+ ( {\sum\limits_{j = {t + 1}}^{T - 1}\quad {( {f + c} )\prod\limits_{j}}} )}}} + {c\prod\limits_{T}}} \}}$

[0097] Next, we need to use iterated expectations to determine thisexpectation. We can write the expectation at time zero as the repeatedexpectation over the various innovations. For example, the equation forinflation can be rewritten as:${E_{0}\quad \exp \{ {{f{\prod\limits_{t}{+ ( {\sum\limits_{j = {t + 1}}^{T - 1}\quad {( {f + c} )\prod\limits_{j}}} )}}} + {c\prod\limits_{T}}} \}} = {{E_{ɛ_{1}}E_{ɛ_{2}}\quad \ldots \quad E_{ɛ_{T}}\quad \exp \{ {{f{\prod\limits_{t}{+ ( {\sum\limits_{j = {t + 1}}^{T - 1}\quad {( {f + c} )\prod\limits_{j}}} )}}} + {c\prod\limits_{T}}} \}} = {E_{ɛ_{1}}E_{ɛ_{2}}\quad \ldots \quad E_{ɛ_{T - 1}}\quad \exp \{ {f{\prod\limits_{t}{+ ( {\sum\limits_{j = {t + 1}}^{T - 1}\quad {( {f + c} )\prod\limits_{j}}} )}}} \} {E_{ɛ_{1}}\lbrack ^{c\prod_{T}} \rbrack}}}$

[0098] Assuming inflation follows a modified square root process:

Π_(t)=μ_(π)+ρ_(π)Π_(t−1)+σ_(π){square root}{square root over(∥Π_(t−1)∥)}ε_(t)

[0099] Where ∥ ∥ denotes the Heaviside function${\prod\limits_{t}} \equiv \{ \begin{matrix}0 & \text{if} & {\prod\limits_{t}{\leq 0}} \\\prod\limits_{t} & \text{if} & {\prod\limits_{t}{> 0}}\end{matrix} $

[0100] Now we recursively start taking the expectations over epsilonstarting at the end and working backward. So: $\begin{matrix}{{E_{ɛ_{T}}\lbrack ^{c\prod_{T}} \rbrack} = \quad {E_{ɛ_{T}}\lbrack ^{{c\quad \mu_{\pi}} + {c\quad \rho_{\pi}{\prod_{T - 1}{{+ c}\quad \sigma_{\pi}\sqrt{\prod_{T - 1}}\quad ɛ_{T}}}}} \rbrack}} \\{\approx \quad ^{c{({\mu_{\pi} + {\rho_{\pi}{\prod_{T - 1}{{{+ 1}/2}c\quad \sigma_{\pi}^{2}\prod_{T - 1}}}}})}}}\end{matrix}$

[0101] Where the approximation is due to the Heaviside function.

[0102] Combining this with the above equation yields:${E_{ɛ_{1}}E_{ɛ_{2}}\quad \ldots \quad E_{ɛ_{T - 1}}\exp \{ {f{\prod\limits_{t}{+ ( {\sum\limits_{j = {t + 1}}^{T - 1}\quad {( {f + c} )\prod\limits_{j}}} )}}} \} {E_{ɛ_{T}}\lbrack ^{c\prod\limits_{t}} \rbrack}} = {E_{ɛ_{1}}E_{ɛ_{2}}\quad \ldots \quad E_{ɛ_{T - 2}}\exp \{ {f{\prod\limits_{t}{+ ( {\sum\limits_{j = {t + 1}}^{T - 1}\quad {( {f + c} )\prod\limits_{j}}} )}}} \} {E_{ɛ_{T - 1}}\lbrack ^{{{c\quad \mu_{\pi}} + {{({{c\quad \rho_{\pi}} + {{1/2}c^{2}\sigma_{\pi}^{2}} + c + f})}\prod_{T - 1}}})} \rbrack}}$

[0103] In general for any time period t, an exponential linear functionof Π has the following expectation: $\begin{matrix}{{E_{ɛ_{t}}\lbrack ^{A_{j} + {B_{j}\prod_{t}}} \rbrack} = \quad {E_{ɛ_{t}}\lbrack ^{A_{j} + {B_{j}{({\mu_{\pi} + {\rho_{\pi}{\prod_{t - 1}{{+ \sigma_{\pi}}{\prod_{t - 1}}ɛ_{t}}}}})}}} \rbrack}} \\{= \quad ^{A_{j} + {B_{j}\mu_{\pi}} + {B_{j}{\prod_{t - 1}{({\rho_{\pi} + {\frac{1}{2}\sigma_{\pi}^{2}B_{j}}})}}}}} \\{= \quad ^{A_{j} + {B_{j}\mu_{\pi}} + {{({B_{j}{({\rho_{\pi} + {\frac{1}{2}\sigma_{\pi}^{2}B_{j}}})}})}\prod_{t - 1}}}} \\{= \quad ^{A_{j - 1} + {B_{j - 1}\prod_{t - 1}}}}\end{matrix}$

[0104] The critical feature is that an exponential linear function of Πremains exponential linear after taking the expectation. This invarianceallows for the backward recursion calculation. Only the constant (A) andthe slope (B) are changing with repeated application of the expectationoperator. The evolution of A and B can be summarized as

A _(J) =A _(J+1)+μ_(π) B _(J+1)

B _(J) =B _(J+1)[ρ_(π)+½σ_(π) ² B _(J+1)]

[0105] In addition, the B_(J) coefficient has to be increased by (c+f)to account for the additional Π_(J) term in the summation. To implementthis recursive algorithm to solve for expected wealth, first define thefollowing indicator variable:${I( {t_{1},t_{2}} )} = \begin{Bmatrix}1 & {{{if}\quad t_{1}} \leq j \leq t_{2}} \\0 & {Otherwise}\end{Bmatrix}$

[0106] Next, the following algorithm may be employed: InitialConditionsJ=T, A_(T)=0, B_(T)=c

[0107] (1) J=J−1

[0108] (2) A_(J)=A_(J+1)+μ_(π)B_(J+1)

[0109] B_(J)=B_(J+1)[ρ_(π)+½σ_(π) ²B_(J+1)]+c·I(t+1, T−1)+f·I(t, T−1)

[0110] (3) if J=0, End

[0111] E(W_(t,T))=e^(A) ^(₁) ^(+B) ^(₁) ^(Π) ^(₀)

[0112] (4) Go To (1)

[0113] The same technique applies to X since it is also a square rootprocess. A similar technique can be used to create a recursive algorithmfor the δ component. The only difference is that δ is an AR(1) processinstead of a square root process.

[0114] In particular,

δ_(t)=μ_(δ)+ρ_(δ)δ_(t−1)+σ_(δ)ε_(t)

[0115] For this AR(1) process, the expectation is of the following form.$\begin{matrix}{{E_{ɛ_{t}}\lbrack ^{A_{j} + {B_{j}\delta_{t}}} \rbrack} = \quad {E_{ɛ_{t}}\lbrack ^{A_{j} + {B_{j}{({\mu_{\delta} + {\rho_{\delta}\delta_{t - 1}} + {\sigma_{\delta}ɛ_{t}}})}}} \rbrack}} \\{= \quad ^{A_{j} + {B_{j}\mu_{\delta}} + {\frac{1}{2}\sigma_{\pi}^{2}B_{j}} + {B_{j}\rho_{\delta}\delta_{t - 1}}}} \\{= \quad ^{A_{j - 1} + {B_{j - 1}\delta_{t - 1}}}}\end{matrix}$

[0116] The evolution of A and B is thus summarized as:

A _(J) =A _(J+1) +B _(J+1)(μ_(δ)+½σ_(δ) ²)

B _(J) =B _(J+1)ρ_(δ)

[0117] The recursive relationship for δ is then:

[0118] InitialConditions J=T, A_(T)=0, B_(T)=d

[0119] (1) J=J−1${(2)\quad A_{J}} = {A_{J + 1} + {B_{J + 1}( {\mu_{\delta} + {\frac{1}{2}\sigma_{\delta}^{2}}} )}}$

[0120] B_(J)=B_(J+1)ρ_(δ)+d·I(t+1, T−1)+g·I(t, T−1)

[0121] (3) if J=0, End

[0122] E (W_(t,T )=e) ^(A) ^(₁) ^(+B) ^(₁) ^(δ) ^(₀)

[0123] (4) Go To (1)

[0124] This framework for calculating expected wealth can also be usedto calculate the variance of wealth for an arbitrary holding period .From the definition of variance, we have:

v ₀(W_(t,T))+E₀({grave over (W)}_(t,T) ²)−E₀(W_(t,T))²

[0125] but $\begin{matrix}{W_{t,T}^{2} = \quad \lbrack {\exp \{ {{\sum\limits_{j = t}^{T - 1}\quad a} + {b\quad X_{j + 1}} + {c{\prod\limits_{j + 1}{{+ d}\quad \delta_{j + 1}}}} + {e\quad X_{j}} + {f{\prod\limits_{j}{{+ g}\quad \delta_{j}}}}} \}} \rbrack^{2}} \\{= \quad {\exp \{ {\sum\limits_{j = t}^{T - 1}\quad {2( {a + {b\quad X_{j + 1}} + {c{\prod\limits_{j + 1}{{+ d}\quad \delta_{j + 1}}}} + {e\quad X_{j}} + {f{\prod\limits_{j}{{+ g}\quad \delta_{j}}}}} )}} \}}}\end{matrix}$

[0126] So the same technique can be used with a simple redefinition ofthe constants to be twice their original values. Similarly, thecovariance between any two core assets can be calculated by simplyadding corresponding constants and repeating the same technique.

[0127] For the current parameter values, the constants for Bills, Bonds,and Equities are: a b c d e F g Bills 0.0077 0 −1 0 1 0.7731 0 Bonds0.0642 −2.5725 −3.8523 0 2.5846 2.9031 0 Equities 0.0331 −2.4062 −3.70694.4431 2.48 2.79 −3.5487

[0128] Above, a methodology was described for calculating core assetanalytic moments for arbitrary horizons. This section describes howthese moments are translated into annualized moments. The proceduredescribed in this section essentially calculates successive annualmoments for a twenty (20) year horizon and computes the arithmeticaverage of these moments. These ‘effective’ annual moments may then beused as inputs into the reverse optimization procedure and theindividual optimization problem.

[0129] For this calculation, first make the following definitions:

[0130] M_(t) ^(J)=Expected return for j^(th) asset over the period t,t+1

[0131] Cov_(t) ^(1,j)=Covariance of returns on asset i with asset j overthe period t, t+1

[0132] These expected returns and covariance are calculated using theformulas described above. The effective annual expected return for assetj is then calculated as:$M^{j} = {\sum\limits_{t = 1}^{T}\quad {\omega_{t}M_{t}^{j}}}$

[0133] Similarly, the effective annual covariance between returns onasset i and returns on asset j are calculated as: (Note, the weights,ω_(t), are between zero and one, and sum to one.)${Cov}^{i,j} = {\sum\limits_{t = 1}^{T}\quad {\omega_{t}{Cov}_{t}^{i,j}}}$

[0134] In one embodiment, this annualizing technique could bepersonalized for a given user's situation. For example, the user'shorizon could specify T, and their level of current wealth and futurecontributions could specify the relevant weights. However for purposesof illustration, the relevant ‘effective’ moments for optimization andsimulation are computed assuming a horizon of 20 years (T=20), and equalweights (i.e. 1/T).

[0135] The techniques described in this section allow for thecalculation of the following effective annual moments: Output parametername Description Units M¹ Bills: expected return Return per year M²Bonds: expected return Return per year M³ Equity: expected return Returnper year Cov^(1,1) Bills: variance of returns (Return per year)²Cov^(2,2) Bonds: variance of returns (Return per year)² Cov^(3,3)Equity: variance of returns (Return per year)² Cov^(1,2) Bills andBonds: covariance (Return per year)² Cov^(1,3) Bills and Equity:covariance (Return per year)² Cov^(2,3) Bonds and Equity: covariance(Return per year)²

[0136] At step 520, a process for increasing diversification isperformed, which is described further below.

[0137] At step 530, a recommended portfolio is output.

Diversification Processing

[0138]FIG. 6 is a flow diagram illustrating diversification processingaccording to one embodiment of the present invention. Conceptually, thediversification processing generally breaks down into an initializationstage, a diversification stage, and an output stage. In the embodimentdepicted, the initialization stage is represented by step 622, thediversification stage includes steps 624, 626, and 628, and the outputstage is represented by step 629. Briefly, after initializing thecandidate portfolio, the diversification stage performs an efficientsearch of an error space for a more diversified portfolio that can beimplemented without exceeding a predetermined diversity budget. Theerror space is an area proximate to or surrounding the initial candidateportfolio and having boundaries defined in terms of expected return,risk, and/or utility, for example.

[0139] At step 622, the candidate portfolio is initialized to theefficient portfolio that was identified in step 510.

[0140] At step 624, a portfolio that is more diversified than thecurrent candidate portfolio is generated. Various approaches forintelligently identifying a more diverse portfolio than the candidateportfolio are described below.

[0141] At step 626, it is determined whether the cost of implementingthe more diversified portfolio is within the diversity budget. If so,then processing continues with step 628; otherwise processing continueswith step 629.

[0142] At step 628, the candidate portfolio is updated with the morediversified portfolio and processing continues with step 624. In thismanner, the most diversified portfolio within the cost constraintsdefined by the diversity budget may be identified.

[0143] At step 629, the current candidate portfolio is output as therecommended portfolio.

[0144] Ultimately, since there might be an extremely large number ofalternative portfolios of financial products to evaluate, one goal ofdiversification processing (step 520) is to limit the diversificationproblem in an intelligent manner. Cost was illustrated above as anexemplary boundary that may act as a stopping condition fordiversification processing. As will be explained with reference to FIG.7, various other conditions may be used to terminate the diversificationprocessing. FIG. 7 is a flow diagram illustrating diversificationprocessing according to another embodiment of the present invention.

[0145] At step 722, the candidate portfolio is initialized to theefficient portfolio that was identified in step 510.

[0146] At step 724, a portfolio that is more diversified than thecurrent candidate portfolio is generated.

[0147] At step 726, the prior candidate portfolio is set to the currentcandidate portfolio and the current candidate portfolio is set to themore diversified portfolio and processing continues with step 728. Inthis manner, depending upon the stopping condition either the portfolioevaluated by the current or prior iteration may be returned as therecommended portfolio depending upon the stopping conditions.

[0148] At step 728, it is determined whether of not one or more stoppingconditions has been achieved. If not, then processing continues withstep 724; otherwise processing continues with step 729. According to oneembodiment, one or more of the following stopping conditions may be usedto terminate the diversification processing:

[0149] (1) the cost exceeds the diversity budget;

[0150] (2) maintaining one or more certain desirable characteristics ofthe initial candidate portfolio constant is no longer feasible;

[0151] (3) the maximum exposure is less than a predetermined minimumexposure threshold;

[0152] (4) exposure to a predetermined minimum or maximum number offinancial products has been achieved;

[0153] (5) a predetermined minimum or maximum number of diversificationiterations has been performed; and

[0154] (6) a predetermined minimum or maximum number of alternateportfolios has been considered.

[0155] At step 729, either the current candidate portfolio or the priorcandidate portfolio is output as the recommended portfolio dependingupon the stopping conditions. For example, if the diversity budget hasbeen exceeded by the current candidate portfolio, then the recommendedportfolio is set to the last candidate that remained within thediversity budget (e.g., the prior candidate portfolio, in this example).However, if a stopping condition other than diversity budget caused theprocessing to terminate, then the recommended portfolio may be set tothe current candidate portfolio. For example, if the condition causingthe diversity processing to terminate was the number of iterations, thenthe recommended portfolio is set to the current portfolio.

Generation of a More Diverse Portfolio

[0156] In addition to defining boundaries of the diversification problemin terms of various combinations of stopping conditions, another goal ofdiversification processing (step 520) is to efficiently search thebounded area (e.g., the error space). FIG. 8 is a flow diagramillustrating the generation of a more diverse portfolio (e.g., steps 624and 724) according to one embodiment of the present invention. Accordingto the embodiment depicted, diversification is achieved by evaluatingadditional alternative optimal portfolios, using Equation #1 and #2, forexample, under various constraints. At step 810, a maximum exposure isselected. The maximum exposure (e.g., UB from above) defines the maximumpercentage of the portfolio's value that may be held in any particularfinancial product for a particular diversification iteration.Importantly, any of a number of approaches may be employed to select themaximum exposure values for iterations of the diversificationprocessing. In one embodiment, the relationship between cost and maximumexposures is assumed to be monotonic. For example, it may be assumed thecost of implementing an efficient portfolio constrained to a maximumexposure of 80% is greater than the cost of implementing an efficientportfolio constrained to a maximum exposure of 90%. In this manner, asearch approach that iteratively lowers the ceiling (as defined by themaximum exposure) to search for a more diverse portfolio may stop once acandidate portfolio exceeds the diversity budget. Similarly, a binarysearch algorithm may be employed that makes use of the monotonicrelationship to select the maximum exposure for the current iteration.

[0157] At step 820, optimization processing is performed subject to oneor more diversity constraints including the maximum exposure for thecurrent iteration. For example, according to one embodiment, risk isheld constant while the maximum exposure constraint is applied.Subsequently, at step 830, one or more characteristics (e.g., expectedreturn, risk, and utility) of the resulting more diversified portfolioare compared to corresponding characteristics of the initiallyidentified optimal portfolio to measure the cost associated with thecurrent level of diversification.

[0158] Having described various approaches to diversificationprocessing, exemplary iterations are now illustrated with reference toFIGS. 9A-9C. FIG. 9A illustrates an initially identified optimalportfolio 950. FIG. 9B illustrates the effect of a maximum exposureconstraint on the portfolio of FIG. 9A; and FIG. 9C illustrates adiversified portfolio after one or more stopping conditions have beenachieved.

[0159] In portfolio 950, financial product 910 represents approximately90% of the portfolio's total value and financial product 920 representsthe remaining 10%. According to this example, in a subsequent iterationillustrated by FIG. 9B, a maximum exposure constraint 941 of 75% isimposed upon the optimization process to arrive at a more diverseportfolio 951. The cost of implementing portfolio 951 as opposed toportfolio 950 is determined to be within the allocated diversity budget;therefore, another iteration may be performed. FIG. 9C represents a morediverse portfolio 952 that results from an even more biting maximumexposure constraint 942. However, the cost, in terms of expected return,risk, and/or utility, of implementing portfolio 952 rather thanportfolio 950 is greater than the diversity budget. Therefore, in thisexample, the recommended portfolio would be portfolio 951 (the mostdiverse candidate portfolio that stayed within the diversity budget).

[0160]FIG. 10 conceptually illustrates an approach for quickly finding adiversified portfolio employing a binary search approach according toone embodiment of the present invention. A maximum exposure 1010 for thefirst iteration is selected. In this example, the maximum exposure 1010for the first iteration is 55% (approximately half way between 100% anda floor 1040 of 10%). If the diversity budget is exceeded in the firstiteration, then in the next iteration the maximum exposure value isselected to be between 100% and 55% where the cost is known to be lower.In the example of FIG. 10, the cost of implementing the candidateportfolio identified by the first iteration is less than the diversitybudget; therefore, the maximum exposure value for the second iteration1020 is selected to be approximately half way between the currentexposure and the floor 1040. Subsequent iterations continue in thismanner by recursively splitting a remaining portion of the maximumexposure range known to meet the budget constraint until one or morestopping conditions are achieved.

Alternative Embodiments

[0161] Many alternative embodiments are contemplated by the inventors ofthe present invention. In the foregoing, the diversification stagesearched for a more diverse portfolio. Alternately, a similar approachcould be used to search for a less diverse portfolio that did not exceeda diversity budget.

[0162] Additionally, expected return was used as an exemplary measure ofthe cost of diversification. Importantly, however, it should beunderstood that the present invention is broadly applicable to portfoliodiversification approaches that use other measurements of cost, such asrisk and/or utility. For example, the expected return on a portfoliocould be held constant, and an efficient search could be performed tofind a more diverse portfolio within a certain risk budget.Alternatively, diversity may be increased until a given utility budgetis exhausted. The utility budget may be defined based upon a userspecific utility function which maps any arbitrary characteristics ofthe portfolio onto a utility measure of desirability, for example. Inother embodiments, the optimization problem can be structured tomaximize an arbitrary measure of diversity subject to an arbitrarybudget.

[0163] Another alternate approach to diversification is to randomly orsystematically search all possible portfolios in the error space for amore diverse portfolio that does not exceed a diversity budget. In oneembodiment, the area of all possible portfolios is randomly searcheduntil a specified stopping condition has occurred. Stopping conditionscould include: exposure to a minimum or maximum number of mutual fundshas been achieved, a predetermined minimum or maximum number ofiterations has been performed, or the search has been performed for aspecified period of time. A cost associated with a portfolio iscalculated by comparing it to the current optimal portfolio. If a newlygenerated portfolio is more desirable than the current optimal portfolio(e.g., the newly generated portfolio is more diverse and the cost iswithin the diversity budget), the current optimal portfolio is replacedwith the newly generated portfolio. When a stopping condition hasoccurred, the most diverse portfolio that does not exceed the diversitybudget is output.

[0164] Certain aspects of the invention described herein have equalapplication to various other optimization problems such as those wherethe inputs into the optimization process are subject to estimation orother types of errors.

[0165] In the foregoing specification, the invention has been describedwith reference to specific embodiments thereof. It will, however, beevident that various modifications and changes may be made theretowithout departing from the broader spirit and scope of the invention.The specification and drawings are, accordingly, to be regarded in anillustrative rather than a restrictive sense.

What is claimed is:
 1. A method comprising: a. determining an initialefficient portfolio of financial products selected by an optimizationprocess from an available set of financial products; b. determining analternate portfolio that is more diverse than the initial efficientportfolio by searching one or more dimensions of an error spaceproximate to or surrounding the initial efficient portfolio for a morediverse portfolio of financial products from the available set offinancial products; c. calculating a cost associated with the alternateportfolio by determining the difference between a characteristic of theinitial efficient portfolio and a corresponding characteristic of thealternate portfolio; and d. selecting the alternate portfolio if thecost is less than or equal to a predetermined diversity budget.
 2. Themethod of claim 1, further comprising repeating b-d if no stoppingconditions are met, wherein said selecting the alternate portfolio alsoconsiders the relative desirability between the alternate portfolio andthe selected alternative portfolio from a previous iteration.
 3. Themethod of claim 1, wherein the stopping conditions comprise one or moreof the following: the cost exceeds the predetermined diversity budget;holding a measure of risk constant is no longer feasible; a maximumexposure is less than a predetermined minimum exposure threshold;exposure to a predetermined maximum number of mutual fund products hasbeen achieved; exposure to a predetermined minimum number of mutual fundproducts has been achieved; a predetermined maximum number of iterationshas been performed; a predetermined minimum number of iterations hasbeen performed; a predetermined maximum number of alternate portfolioshas been considered; and a predetermined minimum number of alternateportfolios has been considered.
 4. The method of claim 3, wherein thepredetermined diversity budget is a default parameter.
 5. The method ofclaim 3, wherein the predetermined diversity budget is a user-specifiedparameter.
 6. The method of claim 1, wherein the determining analternate portfolio further comprises imposing a maximum exposureconstraint that limits holdings in any individual financial product ofthe available set of financial products.
 7. The method of claim 1,wherein the predetermined diversity budget is based at least in partupon a user-specified utility function.
 8. The method of claim 1,wherein the predetermined diversity budget is based at least in partupon a level of investment risk specified by the user.
 9. The method ofclaim 1, wherein the characteristic comprises expected return.
 10. Themethod of claim 1, wherein the characteristic comprises risk.
 11. Themethod of claim 1, wherein the error space is defined in terms of one ormore of expected return, risk, and utility.
 12. The method of claim 1,wherein searching the one or more dimensions of an error space comprisesevaluating portfolios having substantially the same level of risk as theinitial portfolio but having lower expected returns.
 13. The method ofclaim 1, wherein searching one or more dimensions of an error spacecomprises evaluating portfolios having approximately the same expectedreturns as the initial portfolio but having a higher level of risk. 14.The method of claim 1, wherein searching one or more dimensions of anerror space comprises evaluating portfolios with higher diversitylevels, but with utility levels which do not fall below a predeterminedutility floor defined by a utility budget.
 15. A method comprising: a.determining an initial efficient portfolio of mutual fund products froman available set of mutual fund products; b. generating a morediversified portfolio than the initial efficient portfolio from theavailable set of mutual fund products without violating a maximumexposure constraint; c. measuring a cost associated with the morediversified portfolio by comparing a first expected return associatedwith the initial efficient portfolio with a second expected returnassociated with the more diversified portfolio; and d. selecting themore diversified portfolio if the cost associated with the portfolio isless than or equal to a user specified diversity budget.
 16. The methodof claim 15, further comprising modifying the maximum exposureconstraint and repeating b-d if one or more of the stopping conditionsare not met.
 17. The method of claim 15, wherein the stopping conditionscomprise one or more of the following: the cost exceeds thepredetermined diversity budget; holding a measure of risk constant is nolonger feasible; the maximum exposure constraint is less than apredetermined minimum exposure constraint; exposure to a predeterminedmaximum number of mutual fund products has been achieved; exposure to apredetermined minimum number of mutual fund products has been achieved;a predetermined maximum number of iterations has been performed; apredetermined minimum number of iterations has been performed; apredetermined maximum number of alternate portfolios has beenconsidered; a predetermined minimum number of alternate portfolios hasbeen considered; and a specified period of time has expired.
 18. Themethod of claim 15, wherein the maximum exposure constraint represents amaximum exposure to any individual mutual fund in the available set ofmutual funds in terms of a percentage value of the more diversifiedportfolio as a whole.
 19. The method of claim 15, wherein the maximumexposure constraint represents a maximum number of mutual funds that maybe included in the more diversified portfolio.
 20. The method of claim15, wherein the maximum exposure constraint represents a maximumproportion of the more diversified portfolio that may be invested in anyindividual mutual fund of the more diversified portfolio.
 21. The methodof claim 15, wherein the user specified diversity budget is specified inbasis points.
 22. The method of claim 15, wherein the generating a morediversified portfolio comprises searching an error space proximate to orsurrounding the initial efficient portfolio.
 23. The method of claim 22,wherein the generating a more diversified portfolio comprises randomlyselecting a portfolio within the error space.
 24. A method comprising:determining an initial portfolio of financial products from an availableset of financial products, wherein the available set of financialproducts comprise one or more of mutual funds and stocks; determiningone or more alternate portfolios that are more diverse than the initialportfolio; measuring a cost associated with achieving diversity bycomparing one or more characteristics of the initial portfolio and theone or more alternate portfolios; and selecting a portfolio of the oneor more alternate portfolios having an associated cost that is less thanor equal to a predetermined diversity budget.
 25. The method of claim24, wherein the cost is defined in terms of expected return, and whereinthe step of measuring a cost associated with achieving diversitycomprises determining a difference between an expected return associatedwith the initial portfolio and expected returns associated with the oneor more alternate portfolios.
 26. The method of claim 24, wherein thepredetermined diversity budget comprises an annual standard deviationbetween approximately 0 and 0.01.
 27. A method comprising: determiningan initial portfolio and a plurality of more diversified portfolios offinancial products from an available set of financial products;determining a cost associated with each of the plurality of morediversified portfolios, wherein the cost is measured in terms of one ormore of expected returns, risk, and utility; and selecting the mostdiversified portfolio of the more diversified portfolios having anassociated cost that is less than or equal to a predetermined diversitybudget.
 28. The method of claim 27, wherein the cost is defined in termsof risk, and wherein the step of measuring a cost associated withachieving diversity comprises determining a difference between the riskassociated with the initial portfolio and risks associated with the oneor more diversified portfolios.
 29. The method of claim 27, wherein thepredetermined diversity budget is a user specified parameter.
 30. Amethod comprising the steps of: a step for determining an initialportfolio of financial products from an available set of financialproducts; a step for determining one or more alternate portfolios offinancial products from the available set of financial products that aremore diverse than the initial portfolio; a step for measuring a costassociated with achieving diversity based upon one or morecharacteristics of the initial portfolio and the one or more alternateportfolios; and a step for selecting a portfolio of the one or morealternate portfolios having an associated cost of achieving diversitythat is less than or equal to a predetermined diversity budget.
 31. Themethod of claim 30, wherein the step for determining one or morealternate portfolios further comprises a step for imposing a maximumexposure constraint that limits holdings in any individual financialproduct of the available set of financial products to a lesserpercentage than the maximum exposure constraint.
 32. An apparatuscomprising: a portfolio optimization means for simulating portfolioreturn scenarios for one or more portfolios including combinations offinancial products from an available set of financial products; and adiversification processing means comprising: a means for determining aninitial portfolio and a plurality of more diversified portfolios from anavailable set of financial products; a means for determining a costassociated with each of the plurality of more diversified portfolios;and a means for selecting the most diverse portfolio of the morediversified portfolios having an associated cost that is less than orequal to a predetermined diversity budget.
 33. The apparatus of claim32, wherein the cost is defined in terms of a utility, and wherein themeans for determining a cost associated with each of the plurality ofmore diversified portfolios comprises a means for determining adifference between a first utility associated with the initial portfolioand a second utility associated with the plurality of more diversifiedportfolios.
 34. A method comprising: a. determining an initial efficientportfolio of financial products selected by an optimization process froman available set of financial products; b. determining an alternateportfolio by searching one or more dimensions of an error spaceproximate to or surrounding the initial efficient portfolio for aportfolio of financial products from the available set of financialproducts having a predetermined diversity level relative to the initialefficient portfolio; c. calculating a cost associated with the alternateportfolio by comparing the difference between a characteristic of theinitial efficient portfolio and a corresponding characteristic of thealternate portfolio; and d. selecting the alternate portfolio if thecost is less than or equal to a predetermined diversity budget.
 35. Themethod of claim 34, wherein the predetermined diversity level comprisesa higher level of diversity than the initial efficient portfolio. 36.The method of claim 34, wherein the predetermined diversity levelcomprises a lower level of diversity than the initial efficientportfolio.
 37. The method of claim 34, wherein the stopping conditionscomprise one or more of the following: the cost exceeds thepredetermined diversity budget; holding a measure of risk constant is nolonger feasible; a predetermined maximum number of iterations has beenperformed; a predetermined minimum number of iterations has beenperformed; a predetermined maximum number of alternate portfolios hasbeen considered; a predetermined minimum number of alternate portfolioshas been considered; the alternate portfolio comprises a minimum numberof financial products from the available set of financial products andthe cost is less than or equal to the predetermined diversity budget.38. The method of claim 34, wherein the error space is defined in termsof one or more of expected return, risk, and utility.
 39. The method ofclaim 34, wherein searching the one or more dimensions of an error spacecomprises evaluating portfolios having substantially the same level ofrisk as the initial portfolio but having lower expected returns.
 40. Themethod of claim 34, wherein searching one or more dimensions of an errorspace comprises evaluating portfolios having approximately the sameexpected returns as the initial portfolio but having a higher level ofrisk.
 41. The method of claim 34, wherein searching one or moredimensions of an error space comprises evaluating portfolios with higherdiversity levels, but with utility levels which do not fall below apredetermined utility floor defined by a utility budget.